The Hochschild-Kostant-Rosenberg theorem, which relates the Hochschild homology of an algebra to its modules of differential $n$-forms, can be considered a benchmark for smoothness of an algebra. This notion is used here in the search for a conception of smoothness formulated in the context of codifferential categories, both commutative and noncommutative. Since it is desirable to adapt such a conception to the noncommutative context, the theory of this domain is developed considerably; a significant result in this direction establishes a connection between noncommutative codifferential categories and commutative ones. This investigation necessitates, then, both the formulation of a notion of smoothness for $T$-algebras in codifferential categories and an adaptation of the Hochschild-Kostant-Rosenberg theorem to a wide variety of contexts which includes noncommutative ones. The former consideration fosters both the notion of a smooth monad, and a formulation of Andr\'e-Quillen homology in codifferential categories; the latter engenders a highly adaptable version of the Hochschild-Kostant-Rosenberg theorem. Specifically, it is shown that for any algebra modality there exists a corresponding Hochschild-Kostant-Rosenberg theorem. This includes a version of the theorem for the free associative algebra monad, the conclusion of which is satisfied by noncommutative smooth algebras.

The tesis describes software tools used in Category Management process. In the first chapter the theory of Category Management and its impact on todays business is presented. Next chapter describes the relation between information technologies and Category Management. The following chapter introduces software tools for Category Management used by GlaxoSmithKline, s.r.o. The last chapter of the thesis is devoted to case study of cooperation with international retailer Tesco.

The main theme of this thesis is the parallel between results in topos theory and the theory of additive functor categories. In chapter 2, we provide a general overview of the topics used in the rest of the thesis. Locally finitely presentable categories are introduced, and their expression as essentially algebraic categories is explained. The theory of localization for toposes and abelian categories is introduced, and it is shown how these localizations correspond to theories in appropriate logics. In chapter 3, we look at conditions under which the category of modules for a ring object R in a topos E is locally finitely presented, or locally coherent. We show that if E is locally finitely presented, then the category of modules is also; however we show that far stronger conditions are required for the category of modules to be locally coherent. In chapter 4, we show that the Krull-Gabriel dimension of a locally coherent abelian category C is equal to the socle length of the lattice of regular localizations of C. This is used to make an analogous definition of Krull-Gabriel dimension for regular toposes, and the value of this dimension is calculated for the classifying topos of the theory of G-sets, where G is a cyclic group admitting no elements of square order. In chapter 5, we introduce a notion of strong flatness for algebraic categories (in the sense studied by Adamek, Rosickey and Vitale). We show that for a monoid M of finite geometric type, or more generally a small category C with the corresponding condition, the category of M-acts, or more generally the category of set-valued functors on C, has strongly flat covers.

To every small category or Grothendieck topos one may associate its isotropy group, which is an algebraic invariant capturing information about the behaviour of automorphisms. In this thesis, we investigate this invariant in the particular context of quasi-equational theories, which are multi-sorted equational theories in which operations may be partially de fined. It is known that every such theory T has a classifying topos, which is a topos that classi fies all topos-theoretic models of the theory, and that this classifying topos is in fact equivalent to the covariant presheaf category Sets^fpTmod, with fpTmod being the category of all finitely presented, set-based models of T. We then investigate the isotropy group of this classifying topos of T, which will therefore be a presheaf of groups on fpTmod, and show that it encodes a notion of inner automorphism for the theory. The main technical result of this thesis is a syntactic characterization of the isotropy group of a quasi-equational theory, and we illustrate the usefulness of this characterization by applying it to various concrete examples of quasi-equational theories.

Three experiments examined category creation with no feedback and minimal feedback by using modeling to determine number of dimensions subjects attended to. In the first experiment, subjects were shown a series of two-dimensional objects with no training and no feedback and asked to categorize the stimuli. Subjects in experiment 1 mostly attended to one dimension. In the second experiment, subjects shown similar two-dimensional stimuli but were given minimal feedback. Significantly more subjects in experiment 2 attended to both dimensions. In the third experiment, subjects were trained on three related two-dimensional categories and then asked to categorize four. Performance in experiment 3 was similar to that of experiment 1, where subjects mainly attended to 1 dimension. These findings indicate that a more natural feedback structure would help subjects create categories that resemble those used in everyday life.

This thesis describes Category management from the theoretical as well as from the practical point of view. It solves a concrete project between supplier (Danone) and customer (Jednota České Budějovice) on traditional market using all tools conected with Category management in terms of assortment analysis and shelf layouts.

The aim of the thesis is to describe the Czech retail industry and the practice used in the category management. The work compares and evaluates the current state of the ice-cream category management in different retail chains and recommends changes for the next ice-cream season for one of the retailers. The description of the retail industry and the introduction set the background for the thesis and the explanation of the continuous category management provides the framework for the analysis. The empirical part conducts the category planning step by step. The key parts are category assessment and category scorecard, which benchmark different stores in terms of assortment, display, price, and promo. Finally recommendations and conclusions are derived from the analysis to improve the category performance.

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The field of category learning is replete with theories that detail how similarity and comparison based processes are used to learn categories, but these theories are limited to cases in which item and category representations consist of feature vectors. This precludes these methods from learning relational categories, where membership is determined by the structured relations binding the features of a stimulus together. Fortuitously,  researchers within the analogy literature have developed theories of comparison that account for this structure.  This thesis bridges the two approaches, describing a theory of category learning that utilizes the representational frameworks provided by the analogy literature to learn categories that may only be described through the appreciation of the structured relations within their members. This theory is formalized in a model, Building Relations through Instance Driven Gradient Error Shifting (BRIDGES), that shows how relational categories can be learned through attention-driven analogies between concrete exemplars.  This approach is demonstrated through several simulations that compare similarity-based learning and alternatives, such as rule-based abstractions and re-representation.  We then present a series of experiments that explore the reciprocal impact of relational comparison on category structure and category structure on relational comparison.  This work provides a theoretical framework and formal model suggesting that feature-based and relation-based categories are a continuum that are learned through selective attention and similarity-based comparison. / text

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